Waves out of phase: the individual waves combine to give a total amplitude of zero.

6.4.2 Path difference and phase difference

If two sources emit periodic waves in phase, the total amplitude of the disturbance at any point where the waves overlap depends on the phase difference between them.

This phase difference depends on how far each wave has travelled from its source. Waves from the two sources will be in phase provided that the difference in the length of the path is zero or some whole number of wavelengths (A, 2 A, 3A, etc.). The in phase out of phase in phase out of phase

path difference X path difference X2

Figure 6.5 Path difference and phase difference.

waves will be completely out of phase if the path difference is one half wavelength or any odd number of half wavelengths

_, , , etc. , as illustrated in Figure 6.5. ^ 2 2 2 )

6.4.3 When two waves travelling in opposite directions meet

If a transverse pulse is sent down a string tied at one end, it will be reflected and come back upside down. This is according to Newton's third law of motion, which states that action and reaction at the point of reflection are opposite. (The fact that it is reflected upside down is not particularly important as far as the argument that follows is concerned; what is more relevant is that the pulse comes back with the same speed with which it was sent.) If the support is rigid, very little energy is absorbed and the amplitude of the pulse will not be significantly diminished.

incoming pulse / \

V—The reflection of

. a pulse from a

reflected pulse

/ "f rigid boundary.

If, instead of a single pulse, a continuous wave is reflected from a rigid boundary such as a wall, a reflected wave is generated. We have something which seems hard to imagine — two equal waves travelling in opposite directions in the same string.

Figure 6.6 Two waves meet and move out of phase.

The incident and reflected waves will be superimposed on one another, as seen in Figure 6.6. The resultant pattern is shown at three instants in time and tells the story of two waves meeting. The story of two waves meeting. The right-travelling wave is represented by a green line and the left-travelling wave by a red line. The waves are identical and travel with the same speed in opposite directions.

Figure 6.6a is a 'snapshot' taken at the instant when they happen to coincide. The resultant wave at that instant (dotted line) has the same wavelength and twice the amplitude. At the points where the disturbance due to both waves is zero, the resultant is of course also zero. These points are indicated by the dashed lines and are called nodes.

A short time later, in Figure 6.6b, the waves have each moved (the green wave to the right and the red wave to the left). Imagine now that you are sitting at a point in space where there was a node. As time goes on, the displacement due to one of the waves will cause you to go up and that due to the other to go down. For example, at the node, the dashed wave is rising, while the solid wave is falling. But since the wave profiles are symmetrical, the displacements cancel and the resultant remains at zero. And so, by 'sitting on the node' you remain undisturbed! The nodes remain fixed in space.

Halfway between the nodes, the combined waves will give rise to the maximum disturbance and the string will oscillate with an amplitude equal to twice that of the individual waves. These points are called antinodes.

The argument could be stated even more simply on general principles: since the waves moving left and right are identical, there is no preferred direction in which nodes can move, and therefore they will remain where they are.

Figure 6.6c completes the story. The waves have moved on, each by a distance of X/8. At this instant the green and red waves are completely out of phase, and the resultant is zero everywhere. The distance between successive nodes or successive antinodes is X/2.

(The analysis effectively applies only to a string of infinite length. In practice it will only last until the returning wave is again reflected, at the other end, and yet another wave enters the scene.)

6.4.4 A string fixed at both ends

The situation described above is rather unreal, in that we are ignoring what will happen when the reflected wave reaches the other end of the string which must be held by somebody, or attached to something. When the reflected wave comes back to that point it will be reflected again and the wave will continue bouncing back and forth — in principle ad infinitum! Nodes and antinodes will be set up, this time in different places, and the vibrations will soon die out.

If the length of string is equal to some multiple of the internode distance, nodes and antinodes set up by the wave reflected from the far end of the string will be in exactly the same positions as the nodes and antinodes produced by the original wave and reflected wave. In these circumstances the waves combine to give a standing wave.

The particles of the string continue to vibrate in the same sort of way, but with the one important difference that all the particles oscillate in phase at the frequency of the wave, as illustrated in Figure 6.7. The dots show successive positions of string particles at equal intervals of time, indicated by the different shades. Particles situated at antinodes such as A vibrate with maximal amplitude and those at nodes such as B do not vibrate at all. The contemporaneous positions of adjacent particles lie on curves of the same shade. The unique feature of standing waves is that energy is not transmitted but stored as vibrational energy of the particles.

A standing wave stores energy in the oscillations of the particles disturbed by the two waves.

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